# Thread: series 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7...

1. ## series 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7...

Hello!

First, thanks for all the wonderful help during this course of Real Analysis! This website is amazing!

I have to determine whether the series: 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7 + 1/8 + 1/9 - 1/10 - 1/11 ... converges or diverges

I assume I have to do a rearrangement to show this, but I don't know how.

thanks

2. Originally Posted by inthequestofproofs
Hello!

First, thanks for all the wonderful help during this course of Real Analysis! This website is amazing!

I have to determine whether the series: 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7 + 1/8 + 1/9 - 1/10 - 1/11 ... converges or diverges

I assume I have to do a rearrangement to show this, but I don't know how.

thanks
Alternating series test - Wikipedia, the free encyclopedia

3. Okay so that is the sum to infinity of $(-1)^{n+1}\left(\frac{1}{n}\right)$ so look to see if this converges absolutely

4. The series can be written as...

$S= 1 + \sum_{n=1}^{\infty} (-1)^{n} (\frac{1}{2n} + \frac{1}{2n+1})$ (1)

.. so that is an alternating sign series whose terms are decreasing with n and tend to 0 if n tends to infinity: the series converges...

Kind regards

$\chi$ $\sigma$

5. BTW: you can only rearrange the terms if the series converges absolutely.